Maxwell-Equations Theory and Derivation.ppt

Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, electric and magnetic circuits. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.

1. Maxwell Equation
•Maxwell equations are compilation
of basic laws of electromagnetics.
•These basic laws are Gauss law, the
Ampere’s circuital law and the
Faraday’s law.
• These equations describes the world
of electromagnetics.
• Some of these laws were directly
used and some were modified by
Maxwell while compiling the
equations.
James clerk Maxwell
(scottish physicist)

2. Maxwell’s Equations
Basic Law Mathematical Representation
Gauss’ Law for Electrostatics
Gauss’ Law for Magnetism
Faraday’s Law of Induction
Modified Ampere’s circuital
Law
qdSD
s

0
dSB
s
dt
d
dlE
B

dS
t
D
JdlH
s




)(

3. This is the differential form of Gauss law, also called point form
Gauss’s Law for electrostatics
Gauss’s law states that the total electric flux through any closed
surface surrounding charges is equal to the total charges enclosed.
qdSD
s

From Divergence theorem,


Vs
dvdSD 


V
V
dvdvD 
D
Maxwell 1’st Equation
(integral form of gauss law)

4. Gauss Law for magneto statics
The total magnetic flux coming out of a closed surface is equal to the
total magnetic charge(poles) inside the surface.

Maxwell’s 2
nd
Equation
However, magnetic poles always found in pairs.
So, mathematically,
0
dSB
s
(integral form)
By divergence theorem,
0
dvB
V
0B
This is the differential form of gauss law of magneto
statics or also called point form.

5. Maxwell 3
rd
Equation
Faraday’s Law of Electromagnetic Induction
•Biot savart law tells us that the magnetic field is produced by a current.
•Whether the reverse is true i.e. whether the magnetic field would produce
electricity.
• Faraday’s experiments demonstrated that the static magnetic field produce no
current but a time varying magnetic field produce a E.M.F. in a close loop causes
a current to flow.
According to Faraday’s law, the net electromotive force (EMF) in a close loop is
equal to the rate of change of magnetic flux ( ( )enclosed by the loop.
Mathematically,
EMF=

B
dt
d
dlE
B

The negative sign is due to the Lenz’s law.
If the loop has magnetic field density B, the
total flux enclosed by the loop is,
dSB
s
B


6. Maxwell’s 3
rd
Equation cont……
From stoke’s theorem,
t
B
E



This is the Faraday’s law of electromagnetic induction in differential
form or point form
So rate of change of magnetic field
Results electric field





a
l
daB
t
dlE





a
l
daB
t
dlE





aa
daB
t
daE)(
t
H
E


 

7. Maxwell 4
th
Equation
Ampere’s circuital Law
It states that the total magnetic field intensity along the closed loop is equal to
the net current enclosed by the loop.

So, mathematically

From stokes theorem,

JH
This is the differential form of Ampere’s circuit law or point form
IdlH
l

daJdlH
a
l

daJdaH
aa
 
)(

8. Maxwell 4
th
Equation cont…

Taking divergence of the Ampere’s Law of differential
form
We get,

But, from continuity equation
Closed surface having volume charge
density


Consider a closed surface having a volume
charge density . If some charges are
leaving the volume , as a result there is a
current flow from the volume. If the current
density on the surface of the volume is J.


So, Ampere’s Law is not consistent with
continuity equation
JH  )(
0J
t
J





9. Maxwell 4
th
Equation cont…
The net outward current =
Rate of
decrease of
charges
Applying divergence theorem,

( from gauss law)
D
So, in amperes law if we regard as the total current enclosed by the
loop law becomes consistent with the continuity equation.
And the term is called the displacement current density




va
dv
t
daJ 




vv
dv
t
dvJ )( 0



t
J

0)( 



t
D
J
t
D
J



t
D



10. Maxwell 4
th
Equation becomes
Maxwell 4
th
Equation cont…
Differential form or point
form
dS
t
D
JdlH
s




)(
t
D
JH



Integral form
Conduction
current
density
displacement current
density(due to time
varying electric field)


This equation tells that magnetic field exists even in non conducting medium
if electric field is varying with respect to time.
t
E
JH


 
So rate of change of electric field
Results magnetic field